Hamilton–Jacobi–Bellman Equations and Reinforcement Learning: A Theoretical Framework and Empirical Study for Dynamic Credit Decision-Making

Traditional credit scoring models treat lending decisions as static classification, ignoring the dynamic evolution of borrower risk and long-term profit optimisation. This paper reinterprets credit risk management as a discrete-time stochastic optimal control problem and integrates the Hamilton–Jacobi–Bellman (HJB) framework with deep reinforcement learning. Theoretically, we establish the equivalence between a discrete Markov decision process and the HJB equation, prove the existence and uniqueness of the optimal value function, derive the closed-form Riccati solution under linear-quadratic assumptions, and provide a convergence analysis of neural network value iteration. Empirically, using LendingClub loan data (2016–2018), we implement a PPO-based dynamic credit policy. The proposed model achieves an average reward of 1.6726 and a total reward of 867,613, significantly outperforming static baselines as well as a DQN baseline. Ablation experiments show that replacing the policy network with a linear mapping reduces the average reward by 40.8%, confirming the necessity of nonlinear function approximation. Sensitivity analysis and statistical tests (p < 0.001) confirm the robustness and significance of the gains. This work provides a rigorous mathematical foundation and empirical evidence for shifting credit scoring from static classification to dynamic optimisation.